A discrete Morse-based approach to multivariate data analysis

摘要

Multivariate data are becoming more and more popular in several\udapplications, including physics, chemistry, medicine, geography,\udetc. A multivariate dataset is represented by a cell complex and a\udvector-valued function defined on the complex vertices. The major\udchallenge arising when dealing with multivariate data is to obtain\udconcise and effective visualizations. The usability of common visual\udelements (e.g., color, shape, size) deteriorates when the number\udof variables increases. Here, we consider Discrete Morse Theory\ud(DMT) [Forman 1998] for computing a discrete gradient field on a\udmultivariate dataset. We propose a new algorithm, well suited for\udparallel and distribute implementations. We discuss the importance\udof obtaining the discrete gradient as a compact representation of the\udoriginal complex to be involved in the computation of multidimensional\udpersistent homology. Moreover, the discrete gradient field\udthat we obtain is at the basis of a visualization tool for capturing the\udmutual relationships among the different functions of the dataset.